Exotic matter is matter with negative energy density, or to be more precise, where the energy density tensor trace invariant is negative.
There are no known instances of standalone exotic matter in the current universe, but on the other hand, there is no much that we know about the full matter content of the universe, there is a big amount of dark energy driving the expanded acceleration of the universe. It might be due to a cosmological constant, it might be due to quintessence, to negative energy, we really don't know.
We only know that there has been at least once in the lifetime of the universe matter with just the same properties as exotic matter. I'm talking of the inflationary period. This inflation was driven by a field that expanded space (inflaton field, etc. pick the name that suits you)
If we had exotic matter, we would be able to do much more interesting things than an Alcubierre drive. We could build wormholes, or time machines which would consume relatively minimal amounts of the magic stuff
so, someone noticed that a geometry trick allows to create the Alcubierre metric with 50 kilos of unobtainium instead of a Jupiter-mass of the material. Well, big deal, since there is still the little problem of where and how to create the unobtainium
Not trying to diss the work, is still interesting, and when Alcubierre introduced his metric to the world far in the 2000 and something i knew that you would be able to optimize a lot of the metric to get more "sensible" numbers, but my point is that this solves the wrong problem; the problem is that there isn't any unobtainium to test these ideas or run experiments. Until this changes, warp drives, wormholes and time machines are a dead end.